There is no consensus [1] on whether Hilbert's sixth problem: Can physics be axiomatized? has been answered.
David Hilbert proposed a set of pressing concerns for mathematicians in 1900. According to a 2006 report
The sixth problem of the list deals with the axiomatization of physics. It was suggested to Hilbert by his own recent research on the foundations of geometry. He proposed “to treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part. [2]Background on sixth problem
http://www.icm2006.org/proceedings/Vol_III/contents/ICM_Vol_3_82.pdf
Hilbert proposed his problems near the dawn of the Planck revolution, while the debate was raging about statistical methods and entropy, and the atomic hypothesis. It would be another five years before Einstein conclusively proved the existence of atoms.
It would be another year before Russell discovered the set of all sets paradox, which is similar to Cantor's power set paradox. (Though Cantor uncovered this paradox, or perhaps theorem, in the late 1890s, I am uncertain how cognizant of it Hilbert was.)
Interestingly, by the 1920s, Zermelo, Fraenkel and Skolem had axiomatized set theory, specifically forbidding that a set could be an element of itself and hence getting rid of the annoying self-referencing issues that so challenged Russell and Whitehead. But, in the early 1930s, along came Goedel and proved that ZF set theory was either inconsistent or incomplete. His proof actually used Russell's Principia Mathematica as a basis, but it generalizes to apply to all but very limited mathematical systems of deduction. Since mathematical systems can be defined in terms of ZF, it follows that ZF must contain some "true" statements [3] that cannot be tracked back to axioms. So the attempt to axiomatize ZF didn't completely succeed.
In turn, it would seem that Goedel, who began his career as a physicist, had knocked the wind out of Problem 6. Of course, many physicists have not accepted this point, arguing that Goedel's incompleteness theorem applies to only one essentially trivial matter.
In a previous essay, I have discussed the impossibility of modeling the universe as a Turing machine. If that argument is correct, then it would seem that Hilbert's sixth problem has been answered. But I propose here to skip the Turing machine concept and use another idea.
Conceptually, if a number is computable, a Turing machine can compute it. Then again Church's lamda calculus, a recursive method, also allegedly could compute any computable. So are the general Turing machine and the lamda calculus equivalent? Church's thesis conjectures that they are (implying, by the way, that it is unknown whether either misses some computables -- rationals or rational approximations of irrationals.)
But Boolean algebra is the real-world venue used by computer scientists. If an output can't be computed with a Boolean system, no one will bother with it. So it seems appropriate to define an algorithm as anything that can be modeled by an mxn truth table (proof or disproof) and its corresponding Boolean statement.
Specifically, a Boolean circuit can be encoded by a Goedel integer [4]. Similarly, the circuit's corresponding truth table can be Goedel-encoded. Goedel showed (and Rosser helped clarify) that it is logically legitimate to encode some formula P(m,n), within some logical and-or mathematical system S, such that m is the GN (Goedel number) of a formula Q and n is a GN represents the formula "Q has no GN." Which is to say: Q has no proof based on axioms or truth table in S.
So if Q were a theorem in S, S is contradictory. But if ~Q holds, then the conscious observer can see that ~Q is a theorem -- though there is no truth-table or axiomatic verification. That is, S is either inconsistent or incomplete. If we add an axiom to S to obtain a system S' in an attempt to get a complete system, we run into Goedel's conundrum all over again.
Notice that if we wish to build a finite Boolean circuit on the basis of P(m,n), we can't do so -- because Q has no truth table and hence no complete Boolean circuit. What this shows is that logically possible relations cannot all be modeled by computational systems, such as Boolean circuits or Turing machines. In addition, notice the importance of consciousness in grasping the "truth" of a statement even though it is not provable by computation. The mind leaps to a meta-level beyond S. How does it do so? It doesn't add an axiom to get S'; it uses some sort of meta-logic.[5]
The Oxford philosopher John R. Lucas, perhaps anxious to undergird a personal theism [5a], is well known for a paper refuting "mechanism," the notion that the mind operates only via physical causality of a sort that could be modeled by a Turing machine. "Goedel's theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines... almost every mathematical logician I have put the matter to has confessed to similar thoughts, but has felt reluctant to commit himself definitely..."
One way to see Goedel's argument, writes Lucas, is to consider that the possibility that "This formula is unprovable-in-the-system" is false, show that that outcome is impossible, making the formula true, whence it follows that it is unprovable. Lucas notes that even in this form the argument tends to be unpersuasive, but that the point of all Goedel's exacting labor is to prove that there is no "catch" to his theorem, or the fact that the human mind is, in principle, capable of doing something a computer is not capable of.[6]
The philosopher Hilary Putnam, perhaps anxious to undergird his Marxist outlook, was in the other camp, insisting that "all of the issues arise in any computing system capable of answering questions about its own structure, and have nothing to do with the unique nature (if it is unique) of human subjective experience." Putnam, writing before Lucas, however, does not see that specific point, though the philosopher was a widely respected mathematician and logician.[7]
The philosopher Thomas Nagel, an atheist, believes that physics and "neo-Darwinian materialism" are insufficient to give an intelligible representation of the cosmos, writing that "the original scientific revolution itself, which, because of its built-in restrictions, can't result in a 'theory of everything' but must be seen as a stage on the way to a more general form of understanding." [8]
Anyway, not every "truth," and not every legal formula (which are used as a basis of algorithms), can be reduced to axioms. That is, there must always be a formula or "truth" that shows that a "scientific" system of deduction is inconsistent or incomplete.
Now assume that the cosmos has a mechanistic cause-effect structure of the type propounded by physicists. If physics has a complete set of axioms, would it not hold that the cosmos can be modeled by a formal system? And in that case, by the Church-Turing thesis, is it not fair to say that a Boolean circuit could, in theory, represent the cosmic initial conditions, with the cosmic phenomena viewed or inferred by us represented by the output computation?
Obviously, we have to account for quantum uncertainty and other physical "noise." Yet, suppose we can do that. The cosmos being quantized, we can use a digital computer, making a Boolean circuit more justifiable. So in that case the n-character, grammatically correct Boolean statement, or circuit, for the cosmos has a Goedel number, which can be computed by a Boolean circuit (integers being easy enough to compute). But, what circuit computes the cosmic GN? Well, the cosmic Boolean circuit, being all encompassing, must be able to compute the cosmic GN. This is equivalent to saying f(GN) = GN. So we obtain a closed loop. This is equivalent to saying the cosmos is modeled as a computation with no initial conditions.
Because no one can construct a Boolean circuit without initial conditions -- i.e. a specific finite circuit -- it must be that the cosmos cannot be represented by a finite Boolean circuit and hence takes the form P(m,n) in which m is a GN of any finite Boolean circuit and n is a GN that says "some Q has no GN." In other words, some physical truths of the cosmos cannot be tracked back to axioms.
So the Cosmic GN exists if and only if it represents a formula such that there is no axiomatic proof of the cosmos. Hence, logico-mathematical physics cannot be axiomatized. If axiomatization were possible , cosmic physics could be reduced to a Boolean circuit.
Postscript
So, we now face the possibility that two scientific systems of representation may each be correct and yet not equivalent.
To illustrate this idea, consider the base 10 and base 2 number systems. There are some non-integer rationals in base 10 that cannot be expressed in base 2, although approximation can be made as close as we like. These two systems of representation of rationals are not equivalent.
(Cantor's algorithm to compute all the rationals uses the base 10 system. However, he did not show that all base n rationals appear in the base 10 system.)
Apologies for the odd footnote numbering. Use of "control f" should prove helpful.
Comments from a decade ago that I have only just seen.
1. Comment from AL:
Hilbert's Sixth has been solved.
www.EinsteinGravity.com
Conant replies: Are you sure?
3. Comment from Anonymous:
A) Your assertion that ZF must contain some theorems which cannot be proved is rather sloppy. A theorem is a proposition which can be proved. You meant to say that ZF contains some propositions which cannot be proved, and whose negations cannot be proved either. This is incompleteness.
Conant replies: Agreed. ZF contains some apparent truths which cannot be proved without axiomatic extension. I have followed your criticism and clarified the assertion.
B) Goedel's Incompleteness Theorem has nothing to do with whether Physics can be axiomatised. It might have something to do with whether Theoretical Physics can ever be complete. But it might not. After all, Theoretical Physics does not have to include all of ZF. Set theory might go way beyond physical reality, so that Theoretical Physics would not actually contain all of mathematics, and thus Goedel's theorem would not apply. In fact, Goedel also proved a completeness theorem: First order Logic is complete. Now, why would a physicist want second-order quantifiers? So, who knows.
Conant replies: I disagree with you. ZF is an example of the mathematical frame needed for physics. Other frames are possible. But Goedel's result always applies. It doesn't apply to sufficiently weak systems, such as number theory without division. But who can imagine physics resting on such a weak system? Goedel proved that if first order logic is complete, then it contains undecidable propositions.
C) Forget Boolean algorithms and Turing Machines. No algorithm can be exactly modelled by a physical machine: there is always some noise. E.g., no square wave, and hence no string of bits, can be completely reliably produced in the real world, and if it could be produced, the information in it, the «signal», still could not be reliably extracted in an error-free fashion by any finite physical apparatus.
Conant replies: Agreed. Yet the point remains that -- even notionally or theoretically -- a machine model does not suffice to represent the physical activity of the entire cosmos.
2. On the origins of Hilbert’s sixth problem: physics and the empiricist approach to axiomatization by Leo Corry. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society
Abstract of Corry paper:
The sixth of Hilbert’s famous 1900 list of twenty-three problems is a programmatic call for the axiomatization of physical sciences. Contrary to a prevalent view this problem was naturally rooted at the core of Hilbert’s conception of what axiomatization is all about. The axiomatic method embodied in his work on geometry at the turn of the twentieth-century originated in a preoccupation with foundational questions related with empirical science, including geometry and other physical disciplines at a similar level. From all the problems in the list, the sixth is the only one that continually engaged his efforts over a very long period, at least between 1894 and 1932.
4. The method of Goedel numbering
Actual symbol strings are somewhat more sophisticated than the one presented here.
A legal formula P is composed of various symbols from a finite set of elements of the language for S. These symbols are arranged according to a syntax that forbids certain combinations and permits others. So, suppose a subset of symbols makes this formula:
~(x=y).
Let 2 represent "not" or ~. Let 3 represent ( Let 5 represent ) Let 7 represent x Let 11 represent = Let 13 represent yA prime number represents a symbol. The exponent is a positive integer representing the position of the symbol. So the GN for the formula above is 21327311413556 = 104,882,770,060,818,750. By the fundamental theorem of algebra, the order of the symbols is encoded by the factors of a GN.
The fact that Goedel numbers are often enormous and impractical for calculation is of no consequence. What matters is the fact that though any formula can be encoded by a GN, the set of all formulas is not 1-1 with the set of true and false statements.
5. This point has long been a bone of contention among logicians and mathematicians. In The Emperor's New Mind(1986), the mathematician and physicist Roger Penrose championed this view. The reaction was so intense that his next book of a decade later, Shadows of the Mind attempted a detailed rebuttal of the many silly and trivial objections. Yet, some serious logicians felt that he had not adequately made his point.
5a. J.R. Lucas on Knowing God as a Person FREEDOM AND GRACE: ESSAYS (SPCK: 1976), P. 26.
If God is a person, it follows that our relations with God are personal relations. … God is not a Thing. Nor is he an Idea. If we were Platonists, we might believe that there was some technique whereby we could emancipate ourselves from the shackles of our earthly existence and put ourselves on a level with the Forms. But God, being a living spirit, has a different sort of existence from the dead timelessness of the Forms. Knowledge of him is not like knowledge of mathematical truths, which any man can set himself to come to know, but like knowledge of persons, and is essentially an interchange between two parties, requiring not only our wish to know, but his willingness to be known.
6. Lucas's article, Minds, machines and Goedel, first appeared in Philosophy Vol. 36 (1961) and was reprinted in the anthology Minds and Machines (Prentice-Hall 1964), Alan Ross Anderson ed.
7. Putnam's article, Minds and Machines, appeared in Dimensions of Mind: A Symposium (NYU Press 1960), Sydney Hook ed., and was reprinted in the anthology Minds and Machines (Prentice-Hall 1964), Alan Ross Anderson ed.
8. Mind & Cosmos: Why the materialist neo-Darwinian conception of nature is almost certainly wrong (Oxford 2012) by Thomas Nagel.
You are mistaken about Goedel and First-order Logic. Goedel did not prove that first-order logic was undecidable. Only systems complicated enough to contain Peano arithmetic, na$elyan the principle of mathematical induction, turn ouit to be undecidable. And that is because induction is a second-order principle. Actually Prof. Von Plato of Helsinki Univ. Has pointed to the continuing relevance of Gentzen's proof of the consistency of Principia Mathematica (using trenas-finite induction). You should look at von Plato's papers on this.
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